MATH 251, Probability and Statistics I, Fall 2005, Mon. Nov. 7, Day 31After class

Reading:   Continue 6.2.  I think it's very well presented but there are a lot of ideas, and it will take several times through.  First time through: Read to p. 406,  skim "statistical significance" (406-top of 409).  Read "Tests for a population mean" (I'll be doing this early)(409-12), skim the rest.  I'll do the "tests for a population mean"  and then discuss the more general patterns a bit, then the Significance point of view.  When the above is solid, read Statistical Significance (406-9) and 412 on.  Ahead, 6.3, a brief glance at 6.4, then 7.1
Hand in: 

6.54 SSHA--1-sided test of mean 
6.60 Applet--P-value: Also add to your list xbar = -.2, -.4.  Do as instructed; then change the null hypothesis to 2-sided and re-make the list, confirming that your p-values are double the p-values for |x-bar|.  (BTW, the second alternative, generating a sample from the given distribution, has been mis-programmed!  It takes a mean from a sample from the distribution  of X-bars, not the population, so gives wrong answers.)
6.38, 6.39 shading & finding P-values

6.35 stating hypotheses.   State in terms of parameters.
6.34 stating hypotheses.  (For a you have to think of a baseline or control.  State in terms of parameters.
6.33ab wrong statements
6.32 wrong statements

Postpone the rest:
With significance levels
6.40, 41 P to alpha
6.48  bednets very significant (Bednets are now recommended and provided as a first line defense)
6.63, 64  .01 and .05
6.61 onesided z to P to alpha
6.62 two sided z to  P to alpha

 Read, discuss 
 

6.36 stating hypotheses

 Optional 
 

More practice: 
6.54 onesided.
Add your shoebox results to each of  the 2 sheets circulating.  n =4, sigma =4, sigmaxbar =2
    the 4 values || xbar|| z (assuming mean is 20)|| P-value= P(Z > z) || Is P-value < .10?
Add a dot for each of your xbars to the dotplot transparency circulating.

Questions on Confidence Intervals? Day 30

Quiz at end of class.

Significance tests use an elaborate vocabulary, but the basic idea is simple: a result that would "rarely" happen if a claim were true--is good evidence that the claim is NOT true.   Notes Day 30
Start here Wed.
A "Significance level" alpha is a probability level we decide on  in advance as being the "rarely" amount that will push us over into believing (well, sort of) that the H0 claim  is not true. (Historically older language than P-value)
We tend to use simple benchmark numbers for it, like .10 (1 in 10), .05 (1 in 20), .01 (1 in 100).
When the P-value is less  than (or equal to) a particular significance level alpha (say .05), we say,
    "The results are significant at the alpha = .05 level," or "The results are significant (P < .05)" , or "Reject the null hypothesis at level alpha = .05"
A particular scientific discipline may have a commonly accepted set of benchmarks, and language to go with it. (I think I remember .05 = "significant", .01 = "highly significant" in psychology?)
We will be less doctrinaire, use the language "significant at the alpha = ___ level."
(However, "nobody" uses a significance level less rare  than .10, 1 in 10).

Back to lightbulb, H0: µ =1000 hrs.  (Average lightbulb life.)  Competing bulb:   Show it's better.
                            Ha:   µ  > 1000 hrs.  (one-sided)
Get xbar = 1060 hrs, z = 2; P(Z > 2) = P-value =  .0228  More than  2% and less than 3% chance of getting a result this high if we did it again.
                   "Significant at the alpha =.03 level.  Also at the alpha = .05 level"
                    "Not significant at the alpha = .02 level.  Also not significant at the alpha = .01 level"

Lightbulb:  Ha:   µ  Not = 1000 hrs. (two-sided)  Same xbar;  P-value = 2·.0228 = .0456.
                   Our test is just barely significant at the .05 level; it is significant at the .06 level, the .10 level.
                             It's not significant at the .04, .02, .01  level or "higher".
 (It's customary to give the "highest" (smallest) standard significance level that is met, but it's not wrong to say it's significant at any of the "lower" (bigger) levels.)

      In reality, no sharp border between "significance" and "not significant"
               It's better to give the actual P-value if it's known; more information.

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